Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-01T18:44:39.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori

Published online by Cambridge University Press:  20 November 2018

Saeid Azam
Saeid Azam*
Affiliation:
Department of Mathematics, University of Isfahan, P.O. Box 81745, Isfahan, Iran e-mail: saiedazam@yahoo.comazam@sci.ui.ac.ir Institute for Theoretical Physics and Mathematics (IPM)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study them in this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

Keywords

17B6517B6717B40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 2 , 01 April 2006 , pp. 225 - 248
Copyright
Copyright © Canadian Mathematical Society 2006

References

[AABGP] Allison, B., Azam, S., Berman, S., Gao, Y. and Pianzola, A., Extended affine Lie algebras and their root systems. Mem. Amer. Math. Soc. 603(1997), 1–122.Google Scholar
[ABG] Allison, B., Benkart, G. and Gao, Y., Lie tori and extended affine Lie algebras of type BCr, (r ≥ 3). in preparation.Google Scholar
[ABP] Allison, B., Berman, S. and Pianzola, A., Covering Algebras I. Extended affine Lie algebras. J. Algebra 250(2002), 458–516.Google Scholar
[AFY] Allison, B., Faulkner, J. and Yoshii, Y., Lie tori of rank 1. Proceedings of the conference on Lie and Jordan algebras, their representations and applications, Guarujá, Brazil, May 2004, to appear.Google Scholar
[AG] Allison, B. and Gao, Y., The root system and the core of an extended affine Lie algebra. Selecta Math. (N.S.) (2) 7(2001), 149–212.Google Scholar
[AY] Allison, B. and Yoshii, Y., Structurable tori and extended affine Lie algebras of type BC1 . J. Pure Appl. Algebra 184 (2003), no. 2-3, 105–138.Google Scholar
[A1] Azam, S., Extended affine root systems. J. Lie Theory (2) 12(2002).Google Scholar
[A2] Azam, S., Construction of extended affine Lie algebras by the twisting process. Comm. Algebra 28(2000), 2753–2781.Google Scholar
[A3] Azam, S., Nonreduced extended affine Weyl groups. J. Algebra 269(2003), 508–527.Google Scholar
[ABY] Azam, S., Berman, S. and Yousofzadeh, M., Fixed point subalgebras of extended affine Lie algebras. J. Algebra 287(2005), no. 2, 351–380.Google Scholar
[AKY] Azam, S., Khalili, V. and Yousofzadeh, M., Extended affine Lie algebras of type BC. J. Lie Theory, 15(2005), no. 1, 145–181.Google Scholar
[BGK] Berman, S., Gao, Y. and Krylyuk, Y., Quantum tori and the structure of elliptic quasi-simple Lie algebras. J. Funct. Anal 135(1996), 339–386.Google Scholar
[BGKN] Berman, S., Gao, Y., Krylyuk, Y. and Neher, E., The alternative torus and the structure of elliptic quasi-simple Lie algebras of type A2. Trans. Amer.Math. Soc. 347(1995), 4315–4363.Google Scholar
[H-KT] Høegh–Krohn, R. and Torresani, B., Classification and construction of quasi-simple Lie algebras. J. Funct. Anal. 89(1990), 106–136.Google Scholar
[K] Kac, V., Infinite dimensional Lie algebras. third edition, Cambridge University Press, 1990.Google Scholar
[N1] Neher, E., Lie tori. C. R. Math. Rep. Acad. Sci. Canada 26(2004), no. 3, 84–89.Google Scholar
[N2] Neher, E., Extended affine Lie algebras. C. R. Math. Acad. Sci. Soc. R. Can. 26(2004), no. 3, 90–96.Google Scholar
[S] Saito, K., Extended affine root systems 1 (Coxeter transformations) . RIMS, Kyoto Univ. 21(1985) 75–179.Google Scholar
[Y] Yoshii, Y., Coordiate algebras of extended affine Lie algebras of type A1. J. Algebra (1) 234 (2000), 128–168.Google Scholar